Some faces won't merge as if there was another vertice in the middle of the face
You can't do what you're trying to do with this kind of topology. Actually, in 3D, faces are always made of triangles, for example a quad is made of 2 triangles:

If I move up the top right vertex the line between the triangles will rise up, like you would like in your example:

But if I move the top left vertex it will stay flat:

So in your case, you need to add topology, like that:

To join 2 vertices through a face you need to press J rather than F, F will simply create an edge between 2 vertices over the existing face, it won't cut the face.